## Brussels-London

geometry seminar

Day-long geometry seminars with three talks on a common theme

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The seminar involves geometers from the Université libre de Bruxelles and the following London colleges:

University College, King's College, Queen Mary's College, and Imperial College.

#### Brussels-London XXI

# Algebraic geometry

#### International Centre for Mathematical Sciences, Edinburgh - via Zoom, 28th May 2020

The Brussels-London seminar is grateful to the support of the ICMS who have kindly helped us move our seminar series on-line. To attend the seminar you will need to register with the ICMS **before 11am on the day of the seminar**. The ICMS has it's own seminar web-page and you can register here.

Under the new on-line format we have decided to have two talks, rather than three, still on a related theme. Please note that the times are **for the UK timezone.**

#### 13.15 Zoom meeting begins

#### 13.30 Arend Bayer

#### 14.45 Margherita Lelli-Chiesa

**Arend Bayer.** Hyperkähler varieties from Fano varieties via stability conditions.

I will explain how to obtain (families of) Hyperkaehler varieties from (families of) some types of Fano varieties. The construction currently applies to cubic fourfolds, and Gushel-Mukai fourfolds. It goes via moduli spaces of stable objects for stability condition on their "Kuznetsov categories", a certain component of the derived category of these Fano varieties. This, for example, gives two infinite collections of locally complete unirational families of polarised Hyperkaehler varieties. Conversely, I will explain some applications to the geometry of cubic fourfolds.

**Margherita Lelli-Chiesa.** Genus two curves on abelian surfaces.

Let (S,L) be a general (d_1,d_2)-polarized abelian surfaces. The minimal geometric genus of any curve in the linear system |L| is two and there are finitely many curves of such genus. In analogy with Chen's results concerning rational curves on K3 surfaces, it is natural to ask whether all such curves are nodal. In the seminar I will prove that this holds true if and only if d_2 is not divisible by 4. In the cases where d_2 is a multiple of 4, I will construct curves in |L| having a triple, 4-tuple or 6-tuple point, and show that these are the only types of unnodal singularities a genus 2 curve in |L| may acquire. This is joint work with A. L. Knutsen.

#### Registration

To attend the seminar you **must register with the ICMS before 11am on the day of the seminar**.. The ICMS administration will send details of the Zoom meeting to all registered participants on Thursday morning.

#### Past seminars

Click on the seminar names to see the speakers, titles and abstracts.

#### Scientific committee

The organisers of the Brussels-London geometry seminar are Joel Fine (ULB), Lorenzo Foscolo (University College London) and Huy The Nguyen (Queen Mary's University London).

In addition to the organisers, the scientific committee includes:

- Mélanie Bertelson (Université libre de Bruxelles)
- Simone Gutt (Université libre de Bruxelles)
- Jason Lotay (University of Oxford)
- Reto Müller (Queen Mary's College, London)
- Dmitri Panov (King's College, London)
- Bruno Premoselli (Université libre de Bruxelles)
- Felix Schulze (University of Warwick)
- Michael Singer (University College, London)
- Guiseppe Tinaglia (King's College, London)

#### Funding

The seminar is supported by the London Mathematical Society, the ERC and the FNRS. Thanks to this support, there is funding to help graduate students and postdocs attend this seminar. If you are based in London, please contact Felix Schulze. If you are based in Brussels, please contact Joel Fine.

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